Timeline for Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2011 at 16:41 | comment | added | PhotonicCrystal | Fairly good answer is given by G. Emmanuele in: G. Emmanuele, On complemented copies of c0 in spaces of operators II. Comment. Math. Univ. Carolin. 35 (1994), 259-261 | |
| Oct 31, 2011 at 12:14 | comment | added | Philip Brooker | PhotonicCrystal, it would seem to not be of codimension 1, however as a left-ideal it should be maximal in $\mathcal{B}(X)$. | |
| Oct 31, 2011 at 10:25 | comment | added | PhotonicCrystal | Isn't $A$ of codimension 1 in $\mathcal{B}(X)$? | |
| Oct 31, 2011 at 0:50 | comment | added | Philip Brooker | Neat answer. On a tangential point, for $X$ a Hilbert space, reflexive complemented subspaces of $\mathcal{B}(X)$ are isomorphic to Hilbert space(s); I think this was first observed in published form in the early 1990s by Pisier. | |
| Oct 31, 2011 at 0:21 | comment | added | Robert Israel | More generally, any Banach space $X$ is isomorphic to a complemented subspace of ${\cal B}(X)$. Namely for any $x_0 \in X$ and $\phi_0 \in X^*$ with $\phi_0(x_0) = 1$, ${\cal B}(X) = A \oplus B$ where $A = \{T: T(x_0) = 0\}$ and $B = \{x \otimes \phi_0: x \in X\}$. | |
| Oct 30, 2011 at 23:58 | history | answered | Robert Israel | CC BY-SA 3.0 |